Chapter EP, Problem 7.3.10EP

To determine

To calculate: The domain and range of the function $y=5\sqrt{x}+1$ . Also sketch the graph of the function.

Answer to Problem 7.3.10EP

The domain is $\left\{x|x\ge 0\right\}$ range is $\left\{y|y\ge 1\right\}$ . Graph of the function is,

Explanation of Solution

Given information:

The square root function $y=5\sqrt{x}+1$ .

Formula used:

The standard form of square root function is $f\left(x\right)=a\sqrt{x-h}+k$ . Here, h is the horizontal translation, k is the vertical translation and a is the orientation and shape with respect to the function $y=\sqrt{x}$ . The domain is $\left\{x|x\ge h\right\}$ and the range is $\left\{f\left(x\right)|f\left(x\right)\ge k\right\}$ .

Calculation:

Consider the square root function $y=5\sqrt{x}+1$ .

Recall that the standard form of square root function is $f\left(x\right)=a\sqrt{x-h}+k$ . Here, h is the horizontal translation, k is the vertical translation and a is the orientation and shape with respect to the function $y=\sqrt{x}$ . The domain is $\left\{x|x\ge h\right\}$ and the range is $\left\{f\left(x\right)|f\left(x\right)\ge k\right\}$ .

Rewrite the function as,

  $\begin{array}{c}y=5\sqrt{x}+1\\ y=5\sqrt{x-0}+1\end{array}$

The minimum point of the function is at $\left(h,k\right)$ that is $\left(0,1\right)$ .

Next, construct the table of values for $x\ge 0$ .

  $\begin{array}{cc}x& y\\ 0& 1\\ 1& 6\\ 2& 8.1\\ 3& 9.6\\ 4& 11\end{array}$

Graph the coordinates obtained above.

Observe that the shape of the graph is same as that $y=\sqrt{x}$ but it is shifted 1 unit up and vertically expanded.

Also, as the value of x increases the corresponding values of y increases.

Next, the domain is $\left\{x|x\ge 0\right\}$ range is $\left\{y|y\ge 1\right\}$ .